Krein's trace formula for unitary operators and operator Lipschitz functions

TL;DR

This paper characterizes the class of functions on the unit circle for which Krein's trace formula applies to pairs of unitary operators with trace class difference, establishing that these are exactly the operator Lipschitz functions.

Contribution

It proves that the functions satisfying Krein's trace formula for unitary operators are precisely the operator Lipschitz functions, clarifying the functional class involved.

Findings

Krein's trace formula holds for functions on the unit circle that are operator Lipschitz.

The class of functions satisfying the trace formula coincides with operator Lipschitz functions.

Provides a complete characterization of functions for the trace formula in the unitary setting.

Abstract

The main result of the paper is a description of the class of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. We prove that this class of functions coincides with the class of operator Lipschitz functions.